Third level Sierpinski superbuckyball: C20⊗C20⊗C20

As was asked by Bih-Yaw, I think in principle one can always go on the process of using fullerenes to construct superfullerene, and then treat the result as the new module to build a supersuperfullerene etc.. The idea is the same. The problem is always physical limitations: the structure goes too heavy to support itself or you run out of memory trying to build that on a computer. Here is the simplest nontrivial case that I can do on my laptop, a third level superfullerene C20⊗C20⊗C20.

C20⊗C20⊗C20 with g=(1,1): C15920 (Ih)

It is clearer to see when there are only two adjacent nodes:

I would say this is not unbeadable. However, as mentioned, one has to make sure the structure is strong enough to hold itself up. In my experience the best shot is to go with 3mm plastic beads and 0.4mm fish lines, which I'm currently using for the construction of a C60⊗C60 superfullerene. Other possibilities are icosahedron⊗icosahedron⊗C60 or cube⊗cube⊗C60, but I think they are not as representative and illuminating as this one.

Short summary of zome-type superbuckyball part IV: Tori and miscellaneous

Lastly, we will address some other examples of this method of constructing graphitic structures from C60s. We will start with planar tori. All of the zometool models of these tori reported here are composed of entirely blue struts.

Threefold torus⊗C60 with g=1: C216 (C3v)

Fourfold torus⊗C60 with g=0: C224 (C2h)

As noted parenthetically, the rotational symmetry of this structure is actually twofold only.

Fivefold torus⊗C60 with g=0: C280 (C5v)

Sixfold torus⊗C60 with g=0: C336 (D3v)

Hyper cube⊗C60 with g=(0,3,1): C1536 (Th)

Perhaps this hypercube should be also classified as polyhedron in the previous post, since it IS a regular polyhedron in 4D. As can be seen, there are three different kinds of struts (two blues with different lengths and one yellow) in this structure. It is thus more difficult to find a reasonable set of g parameters that suits all of their intricate geometric relations. The CNTs in the outer layer are bent to accommodate this incommensurability. I have to point out that, in graph theory, 4D hypercube cannot be represented by a planar graph. This fact leads to considerable difficulty, if not impossible, in constructing corresponding graphitic structures with out previous approach of using the inner part of TCNTs. However, with the zome-type construction scheme this is nothing different than other graphically simpler structures.

Dodecahedron with V-shape edges⊗C80 with g=1: C4960 (Ih)

Note that one has to at least use C80 (or larger Ih-symmetric fullerenes) instead of C60 for the nodes, since the yellow struts joining at the same node are nearest neighbors to each other. Total fifty C80s were used, twenty for the (inner) dodecahedron and thirty for the (outer) edges. If I make it to the Bridges this year in Enschede, I'll bring a beaded molecule of this model with me.

Last but not least, how can I not play with trefoil knot?

Trefoil knot⊗C60 with g=(0,1,3): C912 (D3)

Unfortunately, so far I have not thought of any general scheme to construct arbitrary torus knots, as trefoil knot is only the simplest nontrivial case of them. In principle as long as the structure (or the space curve as for knots) can be constructed with zometool, there is also graphitic analogs of it and presumably beaded molecules as well. This concludes this series of posts. I'm currently working on a beaded molecule of C60⊗C60 with g=1 posted previously. I'm about half way there and I might talk about some specific beading strategies of this kind of structures later on.

Short summary of zome-type superbuckyball part III: Polyhedra

In this post I will present some other polyhedra built with the same principle. As you might know that the C20 and the C60 discussed in the previous post are exactly regular dodecahedron and the truncated icosahedron (an Archimedean polyhedron).

Cube⊗C60 with g=1: C624

I'd like to note that the symmetry of this structure belongs to the Th point group, although it looks as if it's got a higher symmetry of octahedral group. This is so because of the fact that locally there is only C2 rotational symmetries along each of the joining tubes. And there is no C4 rotational symmetry, not only in this structure but also in all other structures constructed with the golden ratio field where zometool is based on.

Icosahedron⊗C60 with g=2: C1560

I've posted a closely related high-genus structure quite some time ago using a different algorithm. There I treated the construction of high-genus fullerenes by replacing the faces of the underlying polyhedra by some carefully truncated inner part of a toroidal CNT. As suggested by Bih-Yaw that the current scheme of constructing superfullerenes is one another aspect of high-genus fullerene. Previously we are "puncturing holes" along the radial direction and connecting an inner fullerene with an outer one. Here we break and connect fullerenes in the lateral directions. Although topologically they are identical, as you can see the actual shapes of the resulting super-structures are quite different.

For your convenience I repost the structure here for comparison:

The construction of (regular) tetrahedron and octahedron requires the use of green struts. For now I have not come up with the corresponding strategy for green strut yet. We will move on to other polyhedra in the rest of this post.

Small Rhombicosidodecahedron⊗C60 with g=1: C5040

This Archimedean solid is of special interest since the ball of zometool is exactly it. The squares, the equilateral triangles, and the regular pentagons correspond directly to C2, C3, and C5 rotational axes, respectively. The existence of this superfullerene guarantees the possibility of building hierarchy of Sierpinski superbuckyballs. In other words, this superbuckyball can serve as nodes of a "supersuperbuckyball", with the connecting strut automatically defined. Although we are likely to stop at the current (second) level because of physical limitations, either using beads, zometool, or even just computer simulations.

Rhombic triacontahedron⊗C60 with g=1: C3120

You need red struts only for this structure.

Five compound cubes⊗C60 with g=(0,1): C6000

You need blue struts with two different lengths for this structure, which is the reason why the g factor is a two component vector here. Note that at each level the length of the strut (measured from the center of the ball at one end to the center of the other) is inflated by a factor of golden ratio. Thus, comparing to other superfullerenes introduced previously, there is additional strain energy related to the commensurability of the lengths of CNTs. It is always an approximation to use a CNT of certain length to replace the struts of a zometool model. It is also interesting to note that, comparing to the zometool model, this particular superbuckyball makes clear reference to the encompassing dodecahedron. In this perspective it is not surprising that the structure has Ih symmetry.

Dual of C80⊗C60 with g=2: C5880

This structure is obtained by inflating each of the equilateral triangles of a regular icosahedron to four equilateral triangles. An equivalent way of saying this is "inflation with Goldberg vector (2,0)".

In addition to the above mentioned, Dr. George Hart has summarized some of the polyhedra construtable with zometool here. In principle they can all be realized, at least on computers or with beads and threads, by this methodology. And there is going to be one last post in this series to cover those that are not classifiable into categories discussed so far.

Short summary of zome-type superbuckyball part II: Superfullerenes

We will move on to our main and original goal of devising this technique: constructing superfullerenes. Readers familiar with zometool would know that C60 can be built with solely blue struts (two fold rotation axes). So here it is.

C60⊗C60 with g=0: C3240

I've borrowed the notation of Kronecker product (⊗) since these two mathematical operations are in some sense similar: each entry (atom) of the matrix (fullerene) before the ⊗ sign is "expanded" into the second matrix times the original entry (the spatial location of that atom). The meaning of g will become clear once you see an example of g=1 as shown below.

C60⊗C60 with g=1: C4680

It is obvious that g indicates the length of the struts (straight CNTs). In the first case the length is essentially zero, so pairs of heptagons "merge" into octagons at the interface. For clarity the rotatable models of four connected superatoms of the above two superbuckyballs are presented below as well. Since all of these superatoms are identical and can be related through mirror symmetries, readers of interest can start with them to build your own models.

For convenience I also show the rotatable models for the superbuckyball proposed and its beaded model constructed by Bih-Yaw in the previous posts.

C60⊗C60 with g=0: C2700

C60⊗C60 with g=1: C4500

Upon close inspection, notice that there is still local threefold symmetry at each of the node in this case. While on the contrary, there are only mirror symmetries in the zome-type superbuckyball. This asymmetry leads to the fact that there is almost no strain when constructing the beaded model of these structures, or even the actual microscopic realization. This is not an issue concerning the 1D structures in the previous post, since they are all simply connected and there is no such thing as commensurability among multiple struts that join at the same node. However, the above two superstructures seem to be pretty stable and beadable, which surprise me a lot in this regard. According to Bih-Yaw, the tension of the five-member rings and the stress in the six-member rings magically balance each other. This is not so for the dodecahedron case, where tension is built everywhere in the model without being compensated by stress.

Having demonstrated the above mentioned, there is nothing so different in constructing other types of superfullerenes. Below I listed a few that I have done coding with.

C20⊗C60 with g=1: C1560

Although I have not tried to build this one with beads yet, I believe that it is quite doable in the sense of stability as mentioned above.

C80⊗C60 with g=1: C6720

C180⊗C60 with g=1: C15480

I'd like to point out in the last two cases you need red struts as well as blue ones. It can be shown that all (n,n) or (n,0) type icosahedral fullerenes are constructable from zometool (with blue and red struts). For now I just manually find out what are the atoms needing to be deleted/connected, well, in an efficient way. I hope one day I can come up with a general automatic routine that does all these for me, which should be taking account of different orbits in a symmetry group. C180 (a (3,0) Ih fullerene) is the largest one I've ever played with.

I plan to talk about other types of regular polyhedra in the next post.

Short summary of zome-type superbuckyball part I: 1D Linear and Helical C60 Polymers

Recently I've been playing with all kinds of these superbuckyballs, based on the methodology of replacing balls and struts of zometool by Ih-symmetric fullerenes and straight CNTs. Taking C60 for example, blue struts correspond to removing two atoms next to some particular C2 rotation axes. And the connecting CNTs are of chiral vector (4,0). On the other hand, yellow and red struts correspond to C3 and C5 rotation axes, respectively. In addition, due to the property of the golden field, the algebraic field of zometool, superstructures using yellow or red struts only have the possibility of being polymers of C60. This means that the number of atoms is a multiple of 60, i.e. no atoms needed to be deleted or added when constructing the superbuckyball.

Here I will briefly summarize some of the cases I've done coding with. Hopefully I'd soon come up with a short paper ready to submit to the Bridges 2013 on this topic.

First let us start with the trivial C60 dimers. As mentioned, structures with red or yellow struts cases have the atom-preserving property. The C120 isomers corresponding to joining two C60s along their fivefold and threefold axes are shown below. I should mention that they were also discussed in Diudea and Nagy's book . In particular for the C3-joined case there are two possibilities of local atomic connectivity.

C3-fused C120, case 1 (with octagons and pentagons at the interface)

C3-fused C120, case 2 (with heptagons at the interface)

C5-fused C120

For the case of blue struts (twofold rotation axes)

C2-fused C116

C2-fused C132

Some of them were already made previously by us, see here for example. But we did not realize back then this particular connection with zometool. To my knowledge, there has not been any experimental characterization of such dimers. Synthetic chemists do make C60 dimers but those are of partial sp3 characteristics, i.e. some interfacial atoms have four neighbors instead of three. Please refer to Diudea and Nagy's book for further details if you are interested.

One can come up with the one dimensional C60 chains without too much effort by enforcing periodic boundary condition. So the structure repeats itself indefinitely along the direction of polymerization. See for example below.

Also, it is one step away from constructing the 2D analog of this kind of structure.

A little bit more sophisticated extension of the above scheme is to consider helical screw symmetry. A (discrete) helical curve is defined by the angle between adjacent unit cells and the dihedral angle between next-nearest neighbors. I recommend readers of interest to play with the awesome virtual zome program vZome developed by Scott Vorthmann. You have to write Scott an email for the license of the full version of vZome. Anyway, here are some examples of helical C60 polymers.

C3-fused fourfold C60 helix

C5-fused fivefold C60 helix

Notice that if you are looking along the axes of the helices, the C60s that are four/five unit cells away lie exactly on top of each other. Curiously, this result is actually symmetry-determined, since I've tested with the relaxation scheme that does not require such symmetry. In other words, even if I optimize the geometries with full degrees of freedom of a general helix, the screw angles will still be 2*pi/4 or 2*pi/5 in the above cases.

Carbon star and other clover-shaped carbon nanotori

I recently saw a post on the clover-shaped carbon tori by my facebook friend, Tetsuaki Hirata, who is an artist from Himi, Japan and seems to be a frequent visitor of this blog. After I showed Chern about his works, Chern told me he has thought about this kind of clover-shaped carbon nanotori quite some times ago. Indeed, Chern has published a number of papers on the tubular graphitic structures. He is probably one of few people who know a lot about this kind of graphitic structures, especially on how the nonhexagons could influence the structures of a carbon nanotube. I am not surprised that he thought about this kind of structures.

1. Chuang, C.; Fan, Y.-C.; Jin, B.-Y.* Generalized Classification of Toroidal and Helical Carbon Nanotubes J. Chem. Info. Model. 2009, 49, 361-368.
2. Chuang, C; Fan, Y.-C.; Jin, B.-Y.* Dual Space Approach to the Classification of Toroidal Carbon Nanotubes J. Chem. Info. Model. 2009, 49, 1679-1686.
3. Chuang, C; Jin, B.-Y.* Hypothetical toroidal, cylindrical, helical analogs of C60 J. Mol. Graph. Model. 2009, 28, 220-225.
4. Chuang, C.; Fan, Y.-C.; Jin, B.-Y.* On the structural rules of helically coiled carbon nanotubes, J. Mol. Struct. 2012 1008, 1-7.
5. Chuang, C.; Fan, Y.-J.; Jin, B.-Y. Comments on structural types of toroidal carbon nanotubes, arXiv:1212.4567, 2013 submitted to J. Chin. Chem. Soc.

In the first two and the 5th papers, we talked about general structural rules of carbon nanotori and only touched helices briefly. In the next two papers, we discussed very generally how the horizontal and vertical shifts (HS and VS) can be exploited to change the direction of a straight carbon nanotube in order to obtain an arbitrary helically coiled carbon nanotubes. In Chern's Ms thesis, he also showed how to take advantage of HS and VS to create trefoil knots or torus knots in general, which was later summarized in a brief review we wrote, "Systematics of Toroidal, Helically-Coiled Carbon Nanotubes, High-Genus Fullerenes, and Other Exotic Graphitic Materials."  (Procedia Engineering, 2011, 14,  2373-2385).

Clover-shaped TCNTs are just a special class of more general curved carbon nanotubes we considered. A simple strategy is to introduce 180 twists along the tube direction (i.e. 180 degree VS) at suitable positions. I got a few nice figures of clover-shaped TCNTs from Chern the other days.

Among all these clover-shaped tori, I particularly like the five-fold carbon star.